New Approaches to Stability Analysis for Semi-Implicit Numerical Integration Methods

Numerical stability is a critical property of ODE solvers that must be taken into account when choosing a numerical method for solving a certain initial value problem. However, conventional approaches to evaluating numerical stability are not directly applicable to some of the recently reported numerical integration schemes, i.e., semi-implicit or semi-explicit methods, due to their unique right-hand side function calculation principles and existence for systems of order 2 and higher. Furthermore, the stability regions of semi-implicit ODE solvers are found to be significantly influenced by the asymmetry of the simulated system. Thus, the development of specialized techniques is required to address these challenges. In the current study, a complex but straightforward methodology is proposed for the acquisition and graphical representation of the stability regions of semi-implicit integration methods with visualization in three-dimensional space. To cover the deficiency of the stability analysis for recently reported composition methods, we have chosen several composition ODE solvers as test methods. The obtained results supplement our understanding of these algorithms and reveal their strengths and weaknesses.